After introducing the properties of numbers, Spivak states:
"Note, in particular, that $a>0$ if and only if $a$ is in $P$"
I'm not exactly sure how to prove this. The relevant properties are:
"(P10) (Trichotomy law) For every number $a$, one and only one of the following holds:
(i) $a = 0$
(ii) $a$ is in the collection $P$
(iii) $-a$ is in the collection $P$"
and the definition:
"$a>b$ if $a-b$ is in $P$"
Using a previously introduced property and the definition:
$a$ is in $P \implies a>0$
But how do I show:
$a>0 \implies a$ is in $P$?